Mathematical research
Henry Zweiman
Manuscripts in algebra, number theory,
and analysis.
Latest manuscript · P57
Positivity and counterexamples for rising-factorial Toeplitz determinants
We examine the conjectures other than Conjecture 3 in Karp's 2012 paper on rising-factorial Toeplitz determinants. A positive expansion in two-row Toeplitz minors proves Conjecture 1 for all positive shift parameters. A finite-difference and Cauchy--Binet formula proves coefficient nonnegativity for arbitrary determinant order under the corresponding Polya f…
Read the full paperThe manuscripts
57 papers · 5 revised editions- P57
Positivity and counterexamples for rising-factorial Toeplitz determinants
We examine the conjectures other than Conjecture 3 in Karp's 2012 paper on rising-factorial Toeplitz determinants. A positive expansion in two-row Toeplitz minors proves Conjecture 1 for all positive shift parameters. A finite-difference and Cau…
- P56
Negative real zeros of a rising-factorial transform
We prove that a palindromic polynomial with positive constant term and only negative real zeros is carried, by a particular difference of rising-factorial transforms, either to zero or to a polynomial of degree two less with only negative real z…
- P55
A planar comparison argument for radial bound states
We give a planar shooting argument for uniqueness of radial bound states of \Delta u-u+|u|^{p-1}u=0, for each p>1 and each prescribed number of zeros. The argument uses the inverse-branch comparison functionals introduced by Cortázar, García-Hui…
- P54
A spherical Weinstock inequality at large perimeter
For each dimension at least three, we prove that geodesic balls maximize the first nonzero Steklov eigenvalue among smooth geodesically convex domains in the unit sphere with prescribed boundary measure sufficiently close to the area of an equat…
- P53
Endpoint saddles and the local classification of eigenvalue–torsion critical domains
We study the scale-invariant products of the first Dirichlet eigenvalue and a positive power of torsional rigidity, normalized by either volume or perimeter. At each upper local stability threshold identified by Amato, Nitsch, Trombetti and Vill…
- P52
Complete splitting of planar Robin eigenvalues by a single shrinking hole
We study planar Robin eigenvalues when a scaled interior hole shrinks to a prescribed point, with the same fixed nonzero real Robin parameter on the outer and inner boundaries. We establish a complete multiscale description of every branch of a …
- P51
Uniform weak-field and small-flux asymptotics for exterior magnetic Steklov operators
Let a fixed smooth compact obstacle in the plane have connected exterior. We study its magnetic Dirichlet-to-Neumann operator as a constant magnetic field and a nonnegative scalar potential vanish jointly. We obtain a uniform rank-one expansion …
- P50
Heat-content maximization on compact metric graphs at every time
Among finite compact connected metric graphs of prescribed total length with a nonempty Dirichlet vertex set and standard conditions elsewhere, the interval with one Dirichlet and one Neumann endpoint maximizes heat content at every positive tim…
- P49
Sharp stability of the annulus for physical torsion
For the physical torsional rigidity of a planar domain with holes, the concentric annulus maximizes torsion when the outer area and the total hole area are fixed. We prove that the deficit controls the square of the distance of the outer domain …
- P48
Local minimality for the generalized Neumann Hénon problem
We prove strict local minimality, modulo scaling, of radial minimizers for the generalized Neumann Hénon quotient when the exponent of the radial weight is large. The parameter range is determined by the first nonlinear Steklov eigenvalue. It co…
- P47
Sharp total variation bounds from spherical rearrangement
We study the sharp control of total variation by a higher derivative under homogeneous endpoint conditions. For every derivative order n\ge2, we prove that the variation norm on W_0^{n,2}(0,1) is twice the point-evaluation norm, with coincident …
- P46
Ground-state uniqueness on convex domains and sharp logarithmic concavity on balls
We prove uniqueness of the positive Dirichlet ground state of -\Delta u=u\log u^2 on every bounded convex domain, without boundary smoothness assumptions. Every positive log-concave solution satisfies D^2(-\log u)\succ I and is this ground state…
- P45
Spectral localization and uniform reactive capacitance of thin planar patches
We study the Newtonian single-layer operator on planar patches that collapse onto an embedded curve with a variable transverse width. After division by the thickness times its logarithm, the operators converge strongly to transverse averaging fo…
- P44
Robin ground states on convex polyhedra: concavity and vanishing spectral gaps
We disprove the universal positive-Robin fundamental-gap conjecture: in every dimension at least four, for each fixed positive Robin parameter and each fixed diameter, the infimum of the gap over bounded convex polyhedra is zero. An explicit fou…
- P43
A genus obstruction for commuting boundary operators
Let M be a compact connected oriented smooth Riemannian surface with nonempty smooth boundary. We prove that if its Dirichlet-to-Neumann map commutes with the boundary Laplacian, then M has genus zero. This answers both cases of Speciel's Open P…
- P42
Euclidean persistence and sharp stability for anisotropic eigenvalue-torsion products
We optimize the product of the first Dirichlet eigenvalue and a power of torsional rigidity on a fixed Euclidean ball, allowing the anisotropic energy to vary over all normalized symmetric seminorms. For every dimension n\geq2 and every energy e…
- P41
Torsion transplantation and a uniform reverse Kohler-Jobin inequality for Robin problems
We prove that balls maximize the product of the first Robin eigenvalue and a sufficiently large power of the torsional rigidity among bounded planar Lipschitz sets of prescribed area. The exponent threshold is independent of the positive Robin p…
- P40
Boundary selection for low-temperature logarithmic Sobolev constants
We consider Gibbs measures whose convex potential vanishes on a bounded smooth convex domain and grows quadratically in the exterior normal direction. We prove a first-order low-temperature expansion of the sharp logarithmic Sobolev constant. It…
- P39
Counterexamples to a nullity criterion for matrix power maps
For every integer k\ge5 and every field \mathbb F, we construct a diagonal idempotent B\in M_{k+1}(\mathbb F) of nullity two for which the map (X,Y)\mapsto X^k+BY^k is not surjective. An explicit omitted value is the direct sum of three nilpoten…
- P38
manuscript.md
We prove the Amit–Ashurst conjecture for every finite metacyclic p-group, including nonsplit extensions. The proof applies more generally to cyclic extensions of finite modules over commutative chain rings, when conjugation acts as multiplicatio…
- P37
Uniform dominance of short Gorenstein algebras
Let R be a commutative graded Artinian Gorenstein algebra over an arbitrary field, with Hilbert function (1,e,1) and e\ge3. We prove that its dominant index is exactly one: the residue field belongs to level two from every nonzero object of the …
- P36
Dominance and common deformations of short local rings
Let (R,\mathfrak m,k) be a commutative Artinian local ring with \mathfrak m^3=0. We construct a one-dimensional complete Cohen–Macaulay local ring whose quotients by two regular elements outside the square of its maximal ideal are R and \operato…
- P35
Rational points on connected curves cut out by quadrics
Let C be a reduced, geometrically connected projective curve over \mathbb F_q, set-theoretically defined by quadrics. We prove that \C(\mathbb F_q)\le Dq+1 when q\ge3 and D=\deg C\le q+5, with no restriction on the ambient dimension. Consequentl…
- P34
Quasipolynomiality and collision asymptotics for ordinarization counts
Let n_{g,r} count numerical semigroups of genus g and ordinarization number r. For every fixed positive integer r, we prove that n_{g,r} is a quasipolynomial for all positive integers g, answering a question of Cyrusian and Kaplan. Its leading c…
- P33
Sharp completion bounds for polynomial values of matrices
Let \mathbb F be an algebraically closed field of characteristic zero and f\in\mathbb F[t] a nonconstant polynomial. For each \lambda, take the smallest multiplicity of a root of f(t)-\lambda, and let E(f) be the largest of these numbers. We pro…
- P32
Sharp arity bounds for clonoids from mixed-exponent groups
Let p be a prime and s\ge0. We prove that every clonoid from \mathbb Z/p^2\mathbb Z\oplus(\mathbb Z/p\mathbb Z)^s to a module on which p is invertible is generated by its (s+2)-ary functions. The stronger uniform reconstruction identity holds ov…
- P31
A complete classification of a primitive cubic family
We prove that a primitive polynomial X^3+X^2+X+\lambda over \mathbb F_{q^2}, with \lambda primitive in \mathbb F_{q^2}, exists if and only if the characteristic is not three. This determines all surviving cases of a conjecture of Awasthi and Sha…
- P30
manuscript.md
For every odd integer n\geq3, we prove that \nu(C_2^3\oplus C_{2n})=2n+2=\mathsf d(C_2^3\oplus C_{2n}). This disproves Gao’s conjecture that \nu(G)=\mathsf d(G)-1 for every nontrivial finite abelian group. The construction is a zero-sum-free seq…
- P29
manuscript.md
Let G=C_2^a\oplus C_4^b\oplus C_8^c be noncyclic, with c\geq1. We prove that the minimum distance of the signed support of any minimal zero-sum sequence of length \mathsf D(G) divides three. This distance is one if a+c\not\equiv1\pmod3, and it i…
- P28
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We construct a reduced weakly C-monoid with factorial complete integral closure, nonempty conductor, and distance set exactly \{1,2\} which does not satisfy the Structure Theorem for Sets of Lengths. The monoid is the union of a free rank-two bo…
- P27
Conductor ideals of C-monoids with prescribed Krull closure
Let H be a commutative Krull monoid with finite divisor class group. We characterize the ideals of H that occur as conductors of C-monoids with complete integral closure H. For a nonempty proper ideal \mathfrak f, extend its image under a diviso…
- P26
Reconstruction of Krull monoids from conductor submonoids
Let \varphi:H\to F be a divisor theory of a commutative Krull monoid, and let M_\varphi=\{1\}\cup\varphi(H\setminus H^\times)F. We show that M_\varphi determines the reduced monoid of H, together with its divisor theory. The reconstruction takes…
- P25
Primitive idempotents that cannot be homogenized in Koszul algebras
We construct primitive idempotents that no algebra automorphism can carry to a homogeneous idempotent in a locally finite standard graded algebra with semisimple degree zero. The algebra is a matrix algebra over a two-dimensional domain defined …
- P24
The Mendez–Pinto–Villarreal conjecture in unmixed height three
Let I be a monomial ideal whose minimal primary components are irreducible and whose associated primes all have height three. We prove that equality of all symbolic and ordinary powers forces the exponent of each variable to agree in every prima…
- P23
Effective joint Fourier decay at independent Pisot and Salem scales
Let \theta,\phi>1 be multiplicatively independent real algebraic integers, each with all its other conjugates in the closed unit disk. For positive algebraic s,t, we prove an effective upper bound \exp(-c\log\log u/\log\log\log u) for the produc…
- P22
Critical-scale growth bounds for congruence-preserving functions over finite fields
Let R=\mathbb F_q[t]. We prove that a function f:R\to R preserving congruences modulo every irreducible polynomial is a polynomial map if \deg f(A)\le q^{\deg A}/(256q^2) for every input of sufficiently large degree. This replaces the factor q^n…
- P21
Explicit Lie polynomials with prescribed images over finite fields
For every odd prime power q, we give a closed formula in two Lie variables whose image on \mathfrak{sl}_2(\mathbb F_q) is any prescribed conjugation-invariant subset containing zero. The degree is at most 6q-5. The construction converts an arbit…
- P20
The average number of unit-circle zeros in Stanley’s reciprocal polynomial family
For a finite set S of positive integers, put N_S(x)=1-(1-x)\sum_{j\in S}x^j. We determine the asymptotic average number of unit-circle zeros when S is chosen uniformly subject to \max S=2b+1 and reciprocity of N_S. The average is (2b+2)/\sqrt3+o…
- P19
Paley cliques and nonpolynomial counts of totally positive matrices
An element of a finite field is positive if it is a nonzero square, and a matrix is totally positive if all its minors are positive. We determine the number of totally positive two-by-three matrices over every finite field. For a fixed prime p, …
- P18
Integer-polynomial lifting and the classification of algebraic extrapolation sets
For a complex parameter \lambda, let Q_\lambda be the closure of \{0,1\} under the operation (a,b)\mapsto(1-\lambda)a+\lambda b, without taking topological closure. We characterize Q_\lambda for every algebraic integer \lambda\ne0,1 by inequalit…
- P17
A rigid nonassociative shift-associative algebra in dimension five
We construct a five-dimensional complex nilpotent algebra satisfying (ab)c=b(ca) whose orbit is open in the variety of all five-dimensional algebras satisfying this identity. It is nonassociative and has a five-dimensional derivation algebra. It…
- P16
Complement transfer and density bounds for one-sided restricted sumsets
Let A,B\subseteq\mathbb F_p be nonempty, with |B|\le |A|, and forbid at most one pair in each B-column of A\times B. We prove that the remaining sums number at least |A|+|B|-3 whenever |A|+2|B|\le p and |B|\le cp, where c=3520^{-145200}/206. In …
- P15
Countable semiring partitions of real function fields
For every real transcendental number \alpha, we construct a partition of \mathbb Q(\alpha)_{>0} into countably infinitely many nonempty sets closed under addition and multiplication. The construction answers the countable-decomposition question …
- P14
Scalar recurrence sequences with finitely repairable power subsequences
We give a finite algebraic criterion for an integral linear recurrence sequence sampled at n^s to admit a fixed multiplier satisfying the Dold congruences. The criterion permits repeated characteristic roots, root-of-unity quotients, and finite …
- P13
Additive periods of power maps lie in the prime radical
We prove that every additive period of every power map on an associative ring belongs to its prime radical. No identity, characteristic, finiteness, or polynomial-identity assumption is required. The proof combines a substitution using the highe…
- P12
Sharp orders of arithmetical critical groups via canonical lattice simplices
We identify the critical group of an arithmetical structure on a connected simple graph with the vertex-lattice quotient of a canonical lattice simplex. Applying the sharp multiplicity theorem of Averkov, Kasprzyk, Lehmann and Nill, we determine…
- P11
Finite parameter tests for partition congruences and the surviving classes of an elongated partition conjecture
We prove the seven remaining congruence families in a conjecture of Guadalupe on elongated plane partitions, and determine exactly when its final pair of congruences holds. The two progressions conjectured to vanish modulo 3125 vanish identicall…
- P10
Counterexamples to a conjecture on one-sided restricted sumsets
For finite integer sets A,B and a forbidden relation R\subseteq A\times B, let A\mathbin{+_{R}}B be the set of sums represented by pairs outside R. Ouyang conjectured that, when |B|\leq |A| and every vertex of B has forbidden degree at most \Del…
- P09
Unique addition in centerless metabelian Lie algebras of finite length
We prove that a centerless metabelian Lie algebra over a commutative unital ring is a unique-addition Lie ring whenever its underlying module has finite length. Thus every bracket-preserving bijection from it onto an arbitrary Lie ring is additi…
- P08
Componentwise polymatroidality is not preserved by homological shifts
We give a negative answer to a question of Ficarra on homological shifts of componentwise polymatroidal ideals. Over any field, the ideal (x_1,\ldots,x_r,yz,y^a) is componentwise polymatroidal for every r\geq1 and a\geq3, but each of its interme…
- P07
Irreducible compositions in an exceptional unicritical family
We construct irreducible compositions in a positive-shift subfamily left exceptional by a recent theorem on unicritically generated semigroups. Let d\geq4 be even and let a\geq2 be an integer which is not a pth power for any odd prime p\mid d, n…
- P06
Prime support and growth of Stirling repair factors
For fixed k\geq1, let F_k be the least positive integer such that (F_k S(n+k-1,k))_{n\geq1} counts the periodic points of a map, where S(N,k) is a Stirling number of the second kind. We prove math \mathop{\mathrm{rad}}((k-1)!)\mid F_k\mid\mathop…
- P05
Symbolic defect and normalization of whiskered cover ideals
Let J be the cover ideal of the full whiskering of a graph G on n vertices. We express its symbolic defect and integral symbolic defect as differences between the Ehrhart functions of the fractional and ordinary stable-set polytopes and the Hilb…
- P04
Divisors and unique factorizations of Carlitz binomial polynomials
Let A=\mathbb F_q[t]. We classify the divisors in \operatorname{Int}(A) of every finite product of the basic Carlitz binomial polynomials \beta_{q^s}. Each divisor is, up to a unit, a subproduct of the given basic factors. Thus the multiplicativ…
- P03
Maximum-degree local permutation polynomials over the prime field
For every prime power q=p^r>3 and every positive integer n, we prove the existence of a local permutation polynomial over \mathbb F_q with coefficients in \mathbb F_p and total degree n(q-2). The proof uses a subfield patch that preserves the ex…
- P02
Exact Dold repair for matrix powers and integral recurrences
We classify the integral matrices A and positive exponents s for which a fixed positive integer makes every entry of (A^{n^s})_{n\geq1} satisfy the Dold congruences. Such an integer exists exactly when A is semisimple over \mathbb Q and the expo…
- P01
Integral Lefschetz decompositions of fully whiskered simplicial complexes
The square-zero Artinian algebra of a fully whiskered simplicial complex admits an exact integral decomposition, as a module over its Lefschetz operator, into shifted squarefree complete intersections indexed by the faces. This gives an integral…